Editing Dynkin diagram (section)

== Construction from root systems ==

Consider a [[root system]], assumed to be reduced and integral (or "crystallographic"). In many applications, this root system will arise from a [[semisimple Lie algebra]]. Let <math>\Delta</math> be a set of [[Root system#Positive roots and simple roots|positive simple roots]]. We then construct a diagram from <math>\Delta</math> as follows.<ref>{{harvnb|Hall|2015}} Section 8.6</ref> Form a graph with one vertex for each element of <math>\Delta</math>. Then insert edges between each pair of vertices according to the following recipe. If the roots corresponding to the two vertices are orthogonal, there is no edge between the vertices. If the angle between the two roots is 120 degrees, we put one edge between the vertices. If the angle is 135 degrees, we put two edges, and if the angle is 150 degrees, we put three edges. (These four cases exhaust all possible angles between pairs of positive simple roots.<ref>{{harvnb|Hall|2015}} Propositions 8.6 and 8.13</ref>) Finally, if there are any edges between a given pair of vertices, we decorate them with an arrow pointing from the vertex corresponding to the longer root to the vertex corresponding to the shorter one. (The arrow is omitted if the roots have the same length.) Thinking of the arrow as a "greater than" sign makes it clear which way the arrow should go. Dynkin diagrams lead to a [[Root system#Classification of root systems by Dynkin diagrams|classification]] of root systems. The angles and length ratios between roots are [[Root system#Elementary consequences of the root system axioms|related]].<ref>{{harvnb|Hall|2015}} Proposition 8.6</ref> Thus, the edges for non-orthogonal roots may alternatively be described as one edge for a length ratio of 1, two edges for a length ratio of <math>\sqrt{2}</math>, and three edges for a length ratio of <math>\sqrt{3}</math>. (There are no edges when the roots are orthogonal, regardless of the length ratio.)

In the <math>A_2</math> root system, shown at right, the roots labeled <math>\alpha</math> and <math>\beta</math> form a base. Since these two roots are at angle of 120 degrees (with a length ratio of 1), the Dynkin diagram consists of two vertices connected by a single edge: <span class=skin-invert>{{Dynkin|node|3|node}}</span>.